How do you create an infinite ground plane?
For Cycles:

I don't think this really is possible in cycles currently, however you probably don't actually need an infinite plane.

The camera will only render objects up to a finite distance away (end clipping value), so your plane only needs to be larger than this value to appear "infinite".

To do this, enable Limits in Camera settings > Display to display a line visualizing the start and end clipping of the camera:



As long as your end clipping value is reasonably high and your plane is larger than it, the plane will appear reasonably infinite:





If you instead want an effect like in iKlsR's answer, you'll have to do some compositing with renderlayers.


Move your diffuse ground plane to another layer
Create another renderlayer and set the layer to the layer on which your ground plane is, then enable the shadow, AO, and Diffuse indirect passes:



In this example, the ground plane is on layer 2 and the rest of the scene is on layer 1.
Enable Environment on the original renderlayer, which should also be set to only render the layer with all the objects besides the ground plane.
Enable Transparent in Render settings > Film
Composite:




Example .blend